A recently asked question here was solved with the claim that any symmetric square matrix $M$ of the following form is positive definite:
All of the off-diagonal elements are the same positive integer $k$.
Each diagonal element is a positive integer $n_i \gt k$. The diagonal elements may or may not be equal to one another.
The matrix arises as the product of a particular incidence matrix with its transpose. Why is a matrix of this form positive definite?